Lanczos algorithm with matrix product states for dynamical correlation functions

The density-matrix renormalization group (DMRG) algorithm can be adapted to the calculation of dynamical correlation functions in various ways which all represent compromises between computational efficiency and physical accuracy. In this paper we reconsider the oldest approach based on a suitable Lanczos-generated approximate basis and implement it using matrix product states (MPS) for the representation of the basis states. The direct use of matrix product states combined with an ex post reorthogonalization method allows us to avoid several shortcomings of the original approach, namely the multitargeting and the approximate representation of the Hamiltonian inherent in earlier Lanczos-method implementations in the DMRG framework, and to deal with the ghost problem of Lanczos methods, leading to a much better convergence of the spectral weights and poles. We present results for the dynamic spin structure factor of the spin-1/2 antiferromagnetic Heisenberg chain. A comparison to Bethe ansatz results in the thermodynamic limit reveals that the MPS-based Lanczos approach is much more accurate than earlier approaches at minor additional numerical cost.

Figure 1

Comparison of the convergence of the excitation energies and spectral weights for pseudomomentum $k=\pi$ as function of Lanzcos iterations for the spin-$\frac{1}{2}$ Heisenberg chain. The upper panel shows the behavior for the scheme without any reorthogonalization, the lower panel the new scheme. Calculations were done for a chain of length $L=32$ (obc), maximal MPS matrix dimension $M=512$.

Figure 2

The relative error of the first three spectral poles ($k=\pi$, pbc) calculated with the MPS Lanczos method in comparison to the two-spinon excitations from the Bethe ansatz for different system sizes (upper panel) and for $L=32$ for different maximal MPS matrix dimension $M$ (lower panel). The ground state (GS) error is also shown.

Figure 3

Dynamical correlation function for $k=\pi$ for open (upper panel) and periodic boundary (lower panel) conditions. The spectral weights/poles are chosen by the minimal residual scheme.

Figure 4

Rescaled dynamical structure factor $\omega S(k,\omega )$ for $k=\pi$ and periodic boundary conditions. This figure is based on the same data as in Fig. 3.

Figure 5

Rescaled dynamical structure factor $\omega S(k,\omega )$ for $k=\pi$ and open boundary conditions with a Lorentzian broadening of $\eta =0.1$.

Figure 6

Dynamical structure factor $\omega S^{+}(\omega ,k)$ for $k=\pi$ and open boundary conditions with a Lorentzian broadening of $\eta =0.1$, obtained in a standard Lanczos approach (with various numbers of target states $N_L$) and in the correction vector approach; reproduced from Fig. 8 of Ref. 16. Note that due to different schemes of dealing with the open boundary conditions results relate to those in Fig. 5 only up to a proportionality constant.

Figure 7

Dynamical correlation function for $k=\pi /2$ for periodic boundary conditions and $M=512$. The spectral weights/poles are chosen by the minimal residual scheme.